DOCSLIB.ORG

A Subgradient Method for Equilibrium Problems Involving Quasiconvex

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Subgradient Method for Equilibrium Problems Involving Quasiconvex A subgradient method for equilibrium problems involving quasiconvex bifunctions∗ Le Hai Yen†and Le Dung Muu‡ November 4, 2019 Abstract In this paper we propose a subgradient algorithm for solving the equi- librium problem where the bifunction may be quasiconvex with respect to the second variable. The convergence of the algorithm is investigated. A numerical example for a generalized variational inequality problem is provided to demonstrate the behavior of the algorithm. Keywords: Equilibria, Subgradient method, Quasiconvexity. 1 Introduction In this paper we consider the equilibrium problem defined by the Nikaido-Isoda- Fan inequality that is given as Find x∗ ∈ C such that f(x∗,y) ≥ 0 ∀y ∈ C, (EP ) where C is a closed convex set in Rn, f : Rn × Rn → R ∪{+∞} is a bifunction such that f(x, x) = 0 for every (x, x) ∈ C × C ⊂ dom f. The interest of this problem is that it unifies many important problems such as the Kakutani fixed point, variational inequality, optimization, Nash equilibria and some other problems [3, 4, 11, 14] in a convenient way. The inequality in (EP) first was used in [16] for a convex game model. The first result for solution existence arXiv:1911.00181v1 [math.OC] 1 Nov 2019 of (EP) has been obtained by Ky Fan in [6], where the bifunction f can be quasiconvex with respect to the second argument. After the appearance of the paper by Blum-Oettli [4], the problem (EP) is attracted much attention of many authors, and some solution approaches have been developed for solving Problem (EP) see e.g. the monographs [3, 11]. A basic method for solving some classes of Problem (EP) is subgradient (or auxiliary subproblem) one [15, 19], where at each iteration k, having the iterate xk, the next iterate xk+1 is obtained k 1 k 2 by solving the subproblem min{f(x ,y)+ ky − x k : y ∈ C}, with ρk > 2ρk ∗This paper is supported by the NAFOSTED. †Institute of Mathematics, VAST; Email: [email protected] ‡Thang Long University and Institute of Mathematics, VAST; Email: [email protected] 1 0. This subproblem is a strongly convex program whenever f(xk,.) is convex. Since in the case of variational inequality, where f(x, y) := hF (x),y − xi, the k+1 k 1 k iterate x = PC (x − F (x )), the gradient method can be considered 2ρk as an extension of the projection one commonly used in non smooth convex optimization [24] as well as in the variational inequality [7]. The projection algorithms for paramonotone equilibrium problems with f being convex with respect to second variable can be found in [1, 22, 25]. Note that when f(x, .) is quasiconvex the subgradient method fails to apply because of the fact that the objective function f(xk,y)+ 1 ky − xkk2, in general, is neither convex 2ρk nor quasiconvex. To our best knowledge there is no algorithm for equilibrium problems where the bifunction f is quasiconvex with respect to the second variable. In this paper we propose a projection algorithm for solving Problem (EP), where the bifunction may be quasiconvex with respect to the second variable. In order to deal with quasiconvexity we employ the subdifferential for quasiconvex function first introduced in [8], see also [20, 21] for its properties and calculus rules. The subdifferential of a quasiconvex function has been used by some authors for nonsmooth quasiconvex optimization see. e.g. [12, 10, 24], and for quasiconvex feasibility problems [5, 17]. 2 Subdifferentials of quasi-convex functions First of all, let us recall the well known definition of quasiconvex functions, see e.g. [13] Definition 2.1. A function ϕ : Rn → R ∪{+∞} is called quasiconvex on a convex subset C of Rn if and only if for every x, y ∈ C and λ ∈ [0, 1], one has ϕ[(1 − λ)x + λy] ≤ max[ϕ(x), ϕ(y)]. (1) It is easy to see that ϕ is quasiconvex on C if and only if the level set Lϕ(α) := {x ∈ C : ϕ(x) < α}. (2) is convex for any α ∈ R. We consider the following Greenberg-Pierskalla subdifferential introduced in [8] ∂GP ϕ(x) := {g ∈ Rn : hg,y − xi≥ 0 ⇒ ϕ(y) ≥ ϕ(x)}. (3) A variation of the GP-subdifferential is the star-sudifferential that is defined as n ∗ {g ∈ R : hg,y − xi > 0 ⇒ ϕ(y) ≥ ϕ(x)} if x 6∈ D∗ ∂ ϕ(x) := n (R if x ∈ D∗, n n where D∗ is the set of minimizers of ϕ on R . If ϕ is continuous on R , then ∂GP ϕ(x)= ∂∗ϕ(x) ([21]). 2 These subdifferentials are also called quasi-subdifferentials or the normal- subdifferentials. Some calculus rules and optimality conditions for these subd- ifferentials have been studied in [20, 21], among them the following results will be used in the next section. Lemma 2.2. ([10], [20]) Assume that ϕ : Rn → R ∪{+∞} is upper semicon- tinuous and quasiconvex on domϕ. Then ∂GP ϕ(x) 6= ∅ ∀x ∈ domϕ, (4) and ∂∗ϕ(x)= ∂GP ϕ(x) ∪{0}. (5) Lemma 2.3. ([10], [21]) GP GP Rn 0 ∈ ∂ ϕ(x) ⇔ ∂ ϕ(x)= ⇔ x ∈ argmin y∈Rn ϕ(y). (6) The following result follows from Lemma 4 in [10], which can be used to find the subdifferential of a fractional quasiconvex function. Lemma 2.4. ([10]) Suppose ϕ(x)= a(x)/b(x) for all x ∈ domϕ, where a is a convex function, b is finite and positive on domϕ, domϕ is convex and one of the following conditions holds (a) b is affine; (b) a is nonnegative on domϕ and b is concave; Then ϕ is quasiconvex and ∂(a − αb)(x) is a subset of ∂GP ϕ(x) for α = ϕ(x), where ∂ stands for the subdifferential in the sense of convex analysis. 3 Algorithm and its convergence analysis In this section we propose an algorithm for solving Problem (EP), by using the star-sudifferential with respect to the second variable of the bifunction f. As usual, we suppose that C is a nonempty closed convex subset in Rn, f : Rn ×Rn → R∪{+∞} is a bifunction satisfying f(x, x) = 0 and C ×C ⊂ dom f. The solution set of Problem (EP) is denoted by S(EP ). Assumptions (A1) For every x ∈ C, the function f(x, .) is quasiconvex and f(., .) is upper semi continuous on an open set containing C × C; (A2) The bifunction f is pseudomonotone on C, that is f(x, y) ≥ 0 ⇒ f(y, x) ≤ 0 ∀x ∈ C,y ∈ C, and paramonotone on C with respect to S(EP ), that is x ∈ S(EP ),y ∈ C and f(x, y)= f(y, x)=0 ⇒ y ∈ S(EP ). 3 (A3) The solution set S(EP ) is nonempty. Paramonotonicity of bifunctions has been used for equilibrium as well as split equilibrium problems in some papers see e.g. [1, 22, 25, 26]. Various properties of paramonotonicity can be found, for example in [9]. For simplicity of notation. let us denote the sublevel set of the function f(x, .) with value 0 and the star subdifferential of f(x, .) at x as follows Lf (x) := {y : f(x, y) <f(x, x)=0}, ∗ n ∂2 f(x, x) := {g ∈ R |hg,y − xi < 0 ∀y ∈ Lf (x)}. The projection algorithm below can be considered as an extension of the one in [22] to equilibrium problem (EP), where f is quasiconvex with respect to the second variable. Algorithm 3.1. (The Normal-subgradient method) Take a real sequence {αk} satisfying the following conditions αk > 0 ∀k ∈ N, ∞ ∞ 2 k=1 αk =+∞, k=1 αk < +∞. P P Step 0: x0 ∈ C, k =0. Step k: xk ∈ C. k ∗ k k Take g ∈ ∂2 f(x , x ). If gk =0, stop: xk is a solution. Else normalize gk to obtain kgkk =1. Compute k+1 k k x = PC (x − αkg ). If xk+1 = xk then stop. Else update k ←− k +1. Remark 3.2. At each iteration k, in order to check that the iterate xk is k a solution or not one can solve the programming problem miny∈C f(x ,y). In general solving this problem is costly, however in some special cases, for example, when C is a polyhedral convex set and the function f(xk,.) is affine fractional, this program can be solved efficiently by linear programming methods. The following remark ensures the validity of the algorithm. Remark 3.3. (i) Since the star-subdifferential is a cone, one can always nor- malize its nonzero element to obtain an unit vector in the subdifferential. (ii) If Algorithm 3.1 generates a finite sequence, the last point is a solution ∗ k k of (EP). Indeed, if 0 ∈ ∂2 f(x , x ) then by Lemma 2.3, we have k k x ∈ argminy∈Rn f(x ,y), which implies f(xk,y) ≥ 0 for any y ∈ C. 4 k+1 k k k k If x = x then x = PC (x − αkg ). Hence, k k h−αkg ,y − x i≤ 0 ∀y ∈ C, or hgk,y − xki≥ 0 ∀y ∈ C. k ∗ k k GP k k k k k Since g ∈ ∂2 f(x , x ) = ∂2 f(x , x ), we have f(x ,y) ≥ f(x , x ) ≥ 0 for every y ∈ C. For convergence of the algorithm we need the following results that has been proved in [22] for the case the bifunction f is convex in the second variable. Lemma 3.4. The following inequality holds true for every k, k+1 k kx − x k≤ αk. (7) k+1 k k Proof. Since x = PC (x − αkg ), k+1 k k k+1 hx − x + αkg ,y − x i≥ 0 ∀y ∈ C.
Load more

Recommended publications
  • On Variational Inequalities in Semi-Inner Product Spaces
    TAMKANG JOURNAL OF MATHEMATICS Volume 24, Number 1, Spring 1993 ON VARIATIONAL INEQUALITIES IN SEMI-INNER PRODUCT SPACES MUHAMMAD ASLAM NOOR Abstract. In this paper, we consider and study the variational inequal• ities in the setting of semi-inner product spaces. Using the auxiliary principle, we propose and analyze an innovative and novel iterative al• gorithm for finding the approximate solution of variational inequalities. We also discuss the convergence criteria. 1. Introduction Variational inequality theory has been used to study a wide class of free, moving and equilibrium problems arising in various branches of pure and applied science in a general and unified framework. This theory has been extended and generalized in several directions using new and powerful methods that have led to the solution of basic and fundamental problems thought to be inaccessible previously. Much work in this field has been done either in inner product spaces or in Hilbert spaces and it is generally thought that this is desirable, if not essential, for the results to hold. We, in this paper, consider and study the variational inequalities in the setting of semi-inner product spaces, which are more general than the inner product space. It is observed that the projection technique can not be applied to show that the variational inequality problem is Received January 6, 1991; revised March 20, 1992. (1980) AMS(MOS) Subject Classifications: 49J40, 65K05. Keywords and phrases: Variational inequalities, Auxiliary principle, Iterative algorithms, Semi-inner product spaces. 91 92 MUHAMMAD ASLAM NOOR equivalent to a fixed point problem in semi-inner product spaces. This motivates us to use the auxiliary principle technique to suggest and analyze an iterative algorithm to compute the approximate solution of variational inequalities.
    [Show full text]
  • Curriculum Vitae
    Umberto Mosco WPI Harold J. Gay Professor of Mathematics May 18, 2021 Department of Mathematical Sciences Phone: (508) 831-5074, Worcester Polytechnic Institute Fax: (508) 831-5824, Worcester, MA 01609 Email: [email protected] Curriculum Vitae Current position: Harold J. Gay Professor of Mathematics, Worcester Polytechnic Institute, Worcester MA, U.S.A. Languages: English, French, German, Italian (mother language) Specialization: Applied Mathematics Research Interests:: Fractal and Partial Differential Equations, Homog- enization, Finite Elements Methods, Stochastic Optimal Control, Variational Inequalities, Potential Theory, Convex Analysis, Functional Convergence. Twelve Most Relevant Research Articles 1. Time, Space, Similarity. Chapter of the book "New Trends in Differential Equations, Control Theory and Optimization, pp. 261-276, WSPC-World Scientific Publishing Company, Hackenseck, NJ, 2016. 2. Layered fractal fibers and potentials (with M.A.Vivaldi). J. Math. Pures Appl. 103 (2015) pp. 1198-1227. (Received 10.21.2013, Available online 11.4.2014). 3. Vanishing viscosity for fractal sets (with M.A.Vivaldi). Discrete and Con- tinuous Dynamical Systems - Special Volume dedicated to Louis Niren- berg, 28, N. 3, (2010) pp. 1207-1235. 4. Fractal reinforcement of elastic membranes (with M.A.Vivaldi). Arch. Rational Mech. Anal. 194, (2009) pp. 49-74. 5. Gauged Sobolev Inequalities. Applicable Analysis, 86, no. 3 (2007), 367- 402. 6. Invariant field metrics and dynamic scaling on fractals. Phys. Rev. Let- ters, 79, no. 21, Nov. (1997), pp. 4067-4070. 7. Variational fractals. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997) No. 3-4, pp. 683-712. 8. A Saint-Venant type principle for Dirichlet forms on discontinuous media (with M.
    [Show full text]
  • Recent Advances
    RECENT ADVANCES THEORY OF VARIATIONAL INEQUALITIES WITH APPLICATIONS TO PROBLEMS OF FLOW THROUGH POROUS MEDIA J. T. ODEN and N. KIKUCHI The University of Texas at Austin, Austin, TX 78712,U.S.A. TABLE OF CONTENTS PREFACE 1173 INTRODUCTItiN’ : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1174 Introductory comments . 1174 Notations and conventions . 1175 1. VARIATIONAL INEQUALITIES . 1177 1.1 Introduction . 1177 1.2 Some preliminary results . 1180 1.3 Projections in Hilbert spaces . 1183 1.4 The Hartman-Stampacchia theorem . 1187 1.5 Variational inequalities of the second kind ’ : : : : : : : . 1188 1.6 A general theorem on variational inequalities . 1189 1.7 Pseudomonotone variational inequalities of the second kind . 1192 1.8 Quasi-variational inequalities . 1194 1.9 Comments . 1201 2. APPROXIMATION AND NUMERICAL ANALYSIS OF VARIATIONAL INEQUALITIES . 1202 2.1 Convergence of approximations . 1202 2.2 Error estimates for finite element approximations of variational inequalities . 1205 2.3 Solution methods . 1210 2.4 Numerical experiments . 1220 3. APPLICATIONS TO SEEPAGE PROBLEMS FOR HOMOGENEOUS DAMS 1225 3.1 Problem setting and Baiocchi’s transformation . 1225 3.2 A variational formulation . 1229 3.3 Special cases . 1237 4. NON-HOMOGENEOUS DAMS . .......... 1246 4.1 Seepage flow problems in nonhomogeneous dams . .......... 1246 4.2 The case k = k(x) . .......... 1247 4.3 Special cases for k = k(x) . .......... 1249 4.4 The case k = k(y) . : : : : : : : : . .......... 1251 4.5 Special cases for k = k(y) . .......... 1255 4.6 Comments . .......... 1256 5. SEEPAGE FLOW PROBLEMS IN WHICH DISCHARGE IS UNKNOWN .......... 1257 5.1 Dam with an impermeable sheet ............... .......... 1257 5.2 Free surface from a symmetric channel .......... 1267 5.3 A seepage flow problem with a horizontal drain 1 1 : : : : 1 : : : .........
    [Show full text]
  • DPG for Signorini Problem
    ON THE DPG METHOD FOR SIGNORINI PROBLEMS THOMAS FÜHRER, NORBERT HEUER, AND ERNST P. STEPHAN Abstract. We derive and analyze discontinuous Petrov-Galerkin methods with optimal test func- tions for Signorini-type problems as a prototype of a variational inequality of the first kind. We present different symmetric and non-symmetric formulations where optimal test functions are only used for the PDE part of the problem, not the boundary conditions. For the symmetric case and lowest order approximations, we provide a simple a posteriori error estimate. In a second part, we apply our technique to the singularly perturbed case of reaction dominated diffusion. Numerical results show the performance of our method and, in particular, its robustness in the singularly perturbed case. 1. Introduction In this paper we develop a framework to solve contact problems by the discontinuous Petrov- Galerkin method with optimal test functions (DPG method). We consider a simplified model problem ( ∆u + u = f in a bounded domain) with unilateral boundary conditions that resembles a scalar version− of the Signorini problem in linear elasticity [34, 14]. We prove well-posedness of our formulation and quasi-optimal convergence. We then illustrate how the scheme can be adapted to the singularly perturbed case of reaction-dominated diffusion ( "∆u + u = f). Specifically, we use the DPG setting from [22] to design a method that controls the− field variables (u, u, and ∆u) in r the so-called balanced norm (which corresponds to u + "1=4 u + "3=4 ∆u with L2-norm ), cf. [28]. The balanced norm is stronger than the energyk k normkr thatk stems fromk k the Dirichlet bilineark · k form of the problem.
    [Show full text]
  • Full-Text (PDF)
    Mathematical Analysis Vol. 1 (2020), No. 2, 71–80 https:nn maco.lu.ac.ir & Convex Optimization DOI: 10.29252/maco.1.2.7 Research Paper VARIATIONAL INEQUALITIES ON 2-INNER PRODUCT SPACES HEDAYAT FATHI∗ AND SEYED ALIREZA HOSSEINIOUN Abstract. We introduce variational inequality problems on 2-inner product spaces and prove several existence results for variational in- equalities defined on closed convex sets. Also, the relation between variational inequality problems, best approximation problems and fixed point theory is studied. MSC(2010): 58E35; 46B99; 47H10. Keywords: Variational inequality, 2-Inner product spaces, Fixed point theory, Best approximation. 1. introduction The theory of variational inequalities is an important domain of applied mathematics, introduced in the early sixties by Stampacchia and Hartman[10]. It developed rapidly because of its applications. A classical variational in- equality problem is to find a vector u∗ 2 K such that hv − u∗;T (u∗)i ≥ 0; v 2 K; where K ⊆ Rn is nonempty, closed and convex set and T is a mapping from Rn into itself. Later, variational inequality expanded to Hilbert and Banach spaces. So far, a large number of existing conditions have been established. The books [11] and [3] provide a suitable introduction to variational inequal- ity and its applications. For other generalizations of variational inequalities see [14],[15]. The concept of 2-inner product space was introduced by Diminnie, Gahler and A.White[5]. Definition 1.1. Let X be a real linear space with dimX ≥ 2. Suppose that (:; :j:) is a real-valued function defined on X3 satisfying the following conditions: (2I1) (x; xjz) ≥ 0 and (x; xjz) = 0 if and only if x and z are linearly dependent, Date: Received: September 17, 2020, Accepted: December 6, 2020.
    [Show full text]
  • Equivalent Differentiable Optimization Problems and Descent Methods For
    数理解析研究所講究録 第 741 巻 1991 年 147-161 147 Equivalent Differentiable optimization Problems and Descent Methods for Asymmetric Variational Inequality Problems Masao FUKUSHIMA Department of Applied Mathematics and Physics Faculty of Engineering Kyoto University Kyoto 606 Japan Abstract Whether or not the general asymmetric variational inequality prob- lem can be formulated as a differentiable optimization problem has been an open question. This paper gives an affirmative answer to this question. We provide a new optimization problem formulation of the variational inequality problem and show that its objective function is continuously differentiable whenever the mapping involved in the latter problem is continuously differentiable. We also show that under appropriate assumptions on the latter mapping, any stationary point of the optimization problem is a global optimal solution, and hence solves the variational inequality problem. We discuss descent methods for solving the equivalent optimization problem and comment on systems of nonlinear equations and nonlinear complementarity problems. 1 Introduction In this paper we consider the problem of finding a point $x^{*}\in R^{n}$ such that $x^{*}\in S,$ \langle $F(x^{*}),x-x^{*}$ } $\geq 0$ for all $x\in S$, (1.1) 1 148 where $S$ is a nonempty closed convex subset of $R^{n},$ $F$ is a mapping from $R^{n}$ into itself, and $\langle\cdot, \cdot\rangle$ denotes the inner product in $R^{n}$ . This problem is called the variational inequality problem and has been used to study various equilibrium models in economics, operations research, transportation and regional science [2,5,7,13]. In the special case where $S=R_{+}^{n}$ , the nonnegative orthant in $R^{n}$ , problem (1.1) can be rewritten as the nonlinear complementarity problem $x^{*}\geq 0,$ $F(x^{*})\geq 0$ and $(x^{*}, F(x^{*})\rangle$ $=0$ .
    [Show full text]
  • Solving the Signorini Problem on the Basis of Domain Decomposition
    Solving the Signorini Problem on the Basis of Domain Decomp osition Techniques J Sch ob erl Linz Abstract Solving the Signorini Problem on the Basis of Domain Decomp osition Techniques The nite element discretization of the Signorini Problem leads to a large scale constrained minimization problem To improve the convergence rate of the pro jection metho d preconditioning must b e develop ed To b e eective the relative condition numb er of the system matrix with resp ect to the preconditioning matrix has to b e small and the applications of the preconditioner as well as the pro jection onto the set of feasible elements have to b e fast computable In this pap er we show how to construct and analyze such preconditioners on the basis of domain decomp osition techniques The numerical results obtained for the Signorini problem as well as for plane elasticity problems conrm the theoretical analysis quite well AMS Subject Classications T J N F K Key words contact problem variational inequality domain decomp osition precon ditioning Zusammenfassung o sung des Signorini Problems auf der Basis von Gebietszerlegungs Die Au metho den Die Finite Elemente Diskretisierung des SignoriniProblems fuhrt zu einem restringierten Minimierungsproblem mit vielen Freiheitsgraden Zur Verb es serung der Konvergenzrate des Pro jektionsverfahren mussen Vorkonditionierungs techniken entwickelt werden Um ezient zu sein mu die relative Konditionszahl der Systemmatrix in Bezug auf die Vorkonditionierungsmatrix klein sein und die assige Menge Anwendungen der Vorkonditionierung
    [Show full text]
  • Projected Equations, Variational Inequalities, and Temporal Difference Methods
    March 2009; Revised: October 2009 Report LIDS-P-2808 Projected Equations, Variational Inequalities, and Temporal Difference Methods Dimitri P. Bertsekas Laboratory for Information and Decision Systems (LIDS) Massachusetts Institute of Technology MA 02139, USA Email: [email protected] Abstract We consider projected equations for approximate solution of high-dimensional fixed point problems within low- dimensional subspaces. We introduce an analytical framework based on an equivalence with variational inequalities, and a class of iterative algorithms that may be implemented with low-dimensional simulation. These algorithms originated in approximate dynamic programming (DP), where they are collectively known as temporal difference (TD) methods. Even when specialized to DP, our methods include extensions/new versions of TD methods, which offer special implementation advantages and reduced overhead over the standard LSTD and LSPE methods. We discuss deterministic and simulation-based versions of our methods and we show a sharp qualitative distinction between them: the performance of the former is greatly affected by direction and feature scaling, yet the latter asymptotically perform identically, regardless of scaling. 1 I. INTRODUCTION We consider the approximation of a fixed point of a mapping T : n n by solving the projected equation < 7! < x = ΠT (x); (1) where Π denotes projection onto a closed convex subset S^ of n. The projection is with respect to a weighted < 2 1 Euclidean norm , where Ξ is a positive definite symmetric matrix (i.e., x = x0Ξx). We assume that S^ is k · kΞ k kΞ contained in a subspace S spanned by the columns of an n s matrix Φ, which may be viewed as basis functions, × suitably chosen to match the characteristics of the underlying problem: S = Φr r s : (2) f j 2 < g Implicit here is the assumption that s << n, so we are interested in low-dimensional approximations of the high- dimensional fixed point.
    [Show full text]
  • A Hybrid Josephy-Newton Method for Solving Box Constrained Variational Equality Problems Via the D-Gap Function Ji-Ming Peng, Christian Kanzow, Masao Fukushima
    A hybrid Josephy-Newton method for solving box constrained variational equality problems via the D-gap function Ji-Ming Peng, Christian Kanzow, Masao Fukushima To cite this version: Ji-Ming Peng, Christian Kanzow, Masao Fukushima. A hybrid Josephy-Newton method for solving box constrained variational equality problems via the D-gap function. Optimization Methods and Software, Taylor & Francis, 1999, 10 (5), pp.687-710. 10.1080/10556789908805734. hal-01975369 HAL Id: hal-01975369 https://hal.archives-ouvertes.fr/hal-01975369 Submitted on 17 Jan 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A HYBRID JOSEPHY-NEWTON METHOD FOR SOLVING BOX CONSTRAINED VARIATIONAL INEQUALITY PROBLEMS VIA THE D-GAP FUNCTION CHRISTIAN KANZOWb·* JI-MING PENGa,t, t and MASAO FUKUSHIMAc, astate Key Laboratory of Scientific and Engineering Computing , Institute of Computational Mathematics and Scientific/Engineering Computing, Academic Sinica, P.O. Box 2719, Beijing 100080, China; b Institute of Applied Mathematics, University of Hamburg, Bundesstrasse 55, D-20146 Hamburg, Germany; c Department of Applied Mathematics and Physics, Graduate School of Jn.formatics, Kyoto University, Kyoto 606-8501, Japan A box constrained variational inequality problem can be reformulated as an unconstrained minimization problem through the D-gap function.
    [Show full text]
  • Variational Inequalities and Optimization Problems
    Variational Inequalities and Optimization Problems Thesis submitted in accordance with the requirements of the Department of Mathematical Sciences of the University of Liverpool for the degree of Doctor in Philosophy by Yina Liu October 5, 2015 Abstract The main purpose of this thesis is to study weakly sharp solutions of a variational inequality and its dual problem. Based on these, we present finite convergence algorithms for solving a variational inequality problem and its dual problem. We also construct the connection between variational inequalities and engineering problems. We consider a variational inequality problem on a nonempty closed convex subset of Rn. In order to solve this variational inequality problem, we construct the equivalence between the solution set of a variational inequality and optimiza- tion problems by using two gap functions, one is the primal gap function and the other is the dual gap function. We give properties of these two gap functions. We discuss sufficient conditions for the subdifferentiability of the primal gap function of a variational inequality problem. Moreover, we characterize relations between the G^ateaux differentiabilities of primal and dual gap functions. We also build some results for the Lipschitz and locally Lipschitz properties of primal and dual gap functions as well. Afterwards, several sufficient conditions for the relevant mapping to be con- stant on the solution set of a variational inequality has been obtained, including the relations between solution sets of a variational inequality and its dual prob- lem as well as the optimal solution sets to gap functions. Based on these, we characterize weak sharpness of the solution set of a variational inequality by its primal gap function g and its dual gap function G.
    [Show full text]
  • An Evolutionary Variational Inequality Formulation of Supply Chain Networks with Time-Varying Demands
    An Evolutionary Variational Inequality Formulation of Supply Chain Networks with Time-Varying Demands Anna Nagurney and Zugang Liu Department of Finance and Operations Management Isenberg School of Management University of Massachusetts Amherst, Massachusetts 01003 December 2005; updated June 2006 and February 2007 Appears in: Network Science, Nonlinear Science and Dynamic Game Theory Applied to the Study of Infrastructure Systems (2007), pp. 267-302 Terry L. Friesz, Editor, Springer Abstract: This paper first develops a multitiered supply chain network equilibrium model with fixed demands and proves that the governing equilibrium conditions satisfy a finite- dimensional variational inequality. The paper then establishes that the static supply chain network model with its governing equilibrium conditions can be reformulated as a trans- portation network equilibrium model over an appropriately constructed abstract network or supernetwork. This identification provides a new interpretation of equilibrium in supply chain networks with fixed demands in terms of path flows. The equivalence is then further exploited to construct a dynamic supply chain network model with time-varying demands (and flows) using an evolutionary (time-dependent) variational inequality formulation. Re- cent theoretical results in the unification of projected dynamical systems and evolutionary variational inequalities are presented and then applied to formulate dynamic numerical sup- ply chain network examples and to compute the curves of equilibria. An example with step-wise time-dependent demand is also given for illustration purposes. 1 Acknowledgments This research was supported by NSF Grant No. IIS - 002647. The first author also acknowl- edges support from the Radcliffe Institute for Advanced Study at Harvard University under its 2005 – 2006 Felowship Program.
    [Show full text]
  • A Direct Boundary Element Method for Signorini Problems 121
    mathematics of computation volume 55, number 191 july 1990, pages 115-128 A DIRECT BOUNDARYELEMENT METHOD FOR SIGNORINI PROBLEMS HOUDE HAN Abstract. In this paper, a Signorini problem is reduced to a variational in- equality on the boundary, and a direct boundary element method is presented for its solution. Furthermore, error estimates for the approximate solutions of Signorini problems are given. In addition, we show that the Signorini problem may be formulated as a saddle-point problem on the boundary. 1. Introduction The Signorini problems are a class of very important variational inequali- ties, which arise in many practical problems such as the elasticity with unilat- eral conditions [17, 5], the fluid mechanics problems in media with semiper- meable boundaries [4, 7], the electropaint process [1], etc. In the literature, there are many authors who have studied the problems, established the exis- tence and uniqueness, and obtained the regularity results for Signorini problems (see Brézis [3] and Friedman [6]). Furthermore, the numerical solution of the Signorini problem by the finite element method has been discussed [7, 8]. We know that the solution of the Signorini problem satisfies a linear partial differ- ential equation in the domain, even when the problem is nonlinear. Hence, it is natural and advantageous to apply the boundary element method to Signorini problems. An indirect boundary element method for solving Signorini problems has been presented in [9]. In this paper, we will present a direct boundary element method for solving the same problem. First, using the Calderón projector for the traces and the normal derivative of the solution of the Signorini problem, this latter is reduced to a variational inequality on the boundary Y based on a direct boundary el- ement method.
    [Show full text]